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PreprintCC BY 4.0arXiv

Global invertibility of polynomial mappings and connectedness of preimages of large spheres

  • Francisco Braun
  • Zbigniew Jelonek
  • Maria Michalska
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Abstract

We prove that a polynomial mapping ℝ2 → ℝ2 with nowhere zero Jacobian determinant is a global diffeomorphism provided the preimage of large circles are connected. In higher dimensions this result fails and we present families of counterexamples. Yet, in dimension $3$ with codimension of Jelonek set greater than one, we prove that if the preimages of large spheres are connected, then the Jelonek set of the mapping is neither homeomorphic to a line nor can have a single end at infinity.

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Cite

BibTeXRIS
BibTeX
@misc{braun2026global,
  title = {{Global invertibility of polynomial mappings and connectedness of preimages of large spheres}},
  author = {Francisco Braun and Zbigniew Jelonek and Maria Michalska},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.33875},
  eprint = {2610.33875},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.33875},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Global invertibility of polynomial mappings and connectedness of preimages of large spheres
AU  - Francisco Braun
AU  - Zbigniew Jelonek
AU  - Maria Michalska
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.33875
UR  - https://arxiv.org/abs/2610.33875
AB  - We prove that a polynomial mapping ℝ^2 → ℝ^2 with nowhere zero Jacobian determinant is a global diffeomorphism provided the preimage of large circles are connected. In higher dimensions this result fails and we present families of counterexamples. Yet, in dimension $3$ with codimension of Jelonek set greater than one, we prove that if the preimages of large spheres are connected, then the Jelonek set of the mapping is neither homeomorphic to a line nor can have a single end at infinity.
ER  -